1. Entropy = buoyancy
We want to derive the criterion for convective instability in terms of entropy. Convective instability means that if we displace a bubble upward, then
\[\underbrace{\rho^\star_b}_\mathrm{bubble} - \underbrace{\rho^\star}_\mathrm{surroundings} < 0\]
where the \(\star\) superscript indicates the displaced value.
As the buble rises, it expands adiabatically. We’ll write out EOS as \(\rho = \rho(s, p)\). Then the new density inside the bubble is:
\[\rho^\star_b = \rho_b + \frac{d\rho}{dr} \Delta r =
\rho_b +
\left . \frac{\partial \rho}{\partial P} \right |_s
\left . \frac{\partial P} {\partial r} \right |_b \Delta r +
\left . \frac{\partial \rho}{\partial s} \right |_P
\underbrace{\left . \frac{\partial s} {\partial r} \right |_b}_{= 0} \Delta r\]
where \(\partial s/\partial r = 0\) since the bubble expands at constant entropy.
Now the surrounding density in the atmsophere changes as:
\[\rho^\star_\mathrm{sur} = \underbrace{\rho_\mathrm{sur}}_\mathrm{= \rho_b} +
\frac{d\rho}{dr} \Delta r =
\left . \frac{\partial \rho}{\partial P} \right |_s
\left . \frac{\partial P} {\partial r} \right |_\mathrm{sur} \Delta r +
\left . \frac{\partial \rho}{\partial s} \right |_P
\left . \frac{\partial s} {\partial r} \right |_\mathrm{sur} \Delta r\]
So our criteria for instability is:
\[\rho^\star_b - \rho^\star_\mathrm{sur} =
\left . \frac{\partial \rho}{\partial P} \right |_s
\left . \frac{\partial P} {\partial r} \right |_b \Delta r -
\left . \frac{\partial \rho}{\partial P} \right |_s
\left . \frac{\partial P} {\partial r} \right |_\mathrm{sur} \Delta r -
\left . \frac{\partial \rho}{\partial s} \right |_P
\left . \frac{\partial s} {\partial r} \right |_\mathrm{sur} \Delta r < 0\]
But since the bvubble expanded while remaining in pressure equilibrium with its surroundings,
\[\left . \frac{\partial P} {\partial r} \right |_b = \left . \frac{\partial P} {\partial r} \right |_\mathrm{sur}\]
leaving us with:
\[-\left . \frac{\partial \rho}{\partial s} \right |_P
\left . \frac{\partial s} {\partial r} \right |_\mathrm{sur} \Delta r < 0\]
Now our Maxwell relations tell us that
\[\left . \frac{\partial \rho}{\partial s} \right |_P = -\rho^2 \left . \frac{\partial T}{\partial P} \right |_s\]
and our convection criterion becomes:
\[\rho^2 \left . \frac{\partial T}{\partial P} \right |_s
\left . \frac{\partial s} {\partial r} \right |_\mathrm{sur} \Delta r < 0\]
keep in mind that this is for an upward displacement (\(\Delta r > 0\)). This thermodynamic derivative, \(\partial T / \partial P |_s\) is positive for any reasonable EOS (and is related to \(\Gamma_2\)). Therefore, our convective instability criterion becomes:
\[ds < 0\]
Note
This is in the direction of displacement, so this means that high entropy beneath low entropy is unstable to convection.
2. Composition gradients
We want to consider the convective instability in the presence of a composition gradient. The idea is that the composition inside the bubble is fixed, but outside, we have a non-zero \(d\mu/dr\).
As we did in class, we’ll consider the displacement of a bubble upward. Starting with
\(\rho = \rho(P, T, \mu)\), we have
\[ \frac{d\rho}{\rho} =
\underbrace{\left. \left ( \frac{\partial \log \rho}{\partial \log P} \right ) \right |_{T,\mu}}_{\equiv \alpha} \frac{dP}{P} +
\underbrace{\left. \left ( \frac{\partial \log \rho}{\partial \log T} \right ) \right |_{P,\mu}}_{\equiv -\delta} \frac{dT}{T} +
\underbrace{\left. \left ( \frac{\partial \log \rho}{\partial \log \mu} \right ) \right |_{P,T}}_{\equiv \phi} \frac{d\mu}{\mu}
\]
a. general case
Our instability condition is:
\[\Delta \rho = \left [ \left ( \frac{d\rho}{dr} \right )_b - \left ( \frac{d\rho}{dr} \right )_\mathrm{sur} \right ] < 0\]
which is
\[\left [ \left \{ \left ( \frac{\alpha}{P} \frac{dP}{dr} \right )_b -
\left ( \frac{\delta}{T} \frac{dT}{dr} \right )_b\right \} -
\left \{ \left ( \frac{\alpha}{P} \frac{dP}{dr} \right )_\mathrm{sur} -
\left ( \frac{\delta}{T} \frac{dT}{dr} \right )_\mathrm{sur} +
\left ( \frac{\phi}{\mu} \frac{d\mu}{dr} \right )_\mathrm{sur}
\right \} \right ] < 0 \]
and again, since we are in pressure equilibrium, \(dP/dr\) is the same inside and outside of the bubble, so we have:
\[-\left ( \frac{\delta}{T} \frac{dT}{dr} \right )_b +
\left ( \frac{\delta}{T} \frac{dT}{dr} \right )_\mathrm{sur} -
\left ( \frac{\phi}{\mu} \frac{d\mu}{dr} \right )_\mathrm{sur} < 0 \]
Next, we can multiply by the pressure scale height, \(-P \frac{dr}{dP} > 0\), and get:
\[\delta \left . \frac{d\log T}{d\log P} \right |_b
-\delta \left . \frac{d\log T}{d\log P} \right |_\mathrm{sur} +
\phi \left .\frac{d\log\mu}{d\log P} \right |_\mathrm{sur} < 0 \]
and assuming that \(\delta > 0\), we have:
\[\left . \frac{d\log T}{d\log P} \right |_\mathrm{sur} >
\underbrace{\left . \frac{d\log T}{d\log P} \right |_\mathrm{ad}}_\mathrm{bubble~is~ adiabatic} +
\frac{\phi}{\delta} \left .\frac{d\log\mu}{d\log P} \right |_\mathrm{sur}\]
This is the same as:
\[\nabla > \nabla_\mathrm{ad} + \frac{\phi}{\delta} \nabla_\mu\]
Note
Usually \(\nabla \mu > 0\) (heavier composition inwards), so the composition gradient stabilizes the atmosphere against convection.
b. Ideal gas + radiation
Now consider:
\[P = \frac{\rho k T}{\mu m_u} + \frac{1}{3} a T^4\]
Expressing this as \(\rho = \rho(P, T, \mu)\), we have:
\[\rho = \left (P - \frac{1}{3} a T^4 \right ) \frac{\mu m_u}{k T}\]
then:
\[\left . \frac{\partial \rho}{\partial \mu} \right |_{P,T} = \frac{\rho}{\mu}\]
\[\left . \frac{\partial \rho}{\partial T} \right |_{P, \mu} = -\frac{\rho}{T} \left [ 1 + 4 \frac{1 - \beta}{\beta} \right ]\]
where \(\beta = P_g / P\).
Then we can compute the indices in our expansion:
\[-\delta = \left . \frac{\partial \log \rho}{\partial \log T} \right |_{P,\mu} = 1 + 4 \frac{1-\beta}{\beta}\]
\[\phi = \left . \frac{\partial \log \rho}{\partial \log \mu} \right |_{P,T} = 1\]
and our convective instability criterion becomes:
\[\nabla > \nabla_\mathrm{ad} + \frac{\beta}{4 - 3\beta} \nabla_\mu\]